1. Hydrostatics Dp Dz
Air
Parcel 5 1 2 3 4 6 7 8 9 g

2. Reading Hess Wallace & Hobbs Bohren & Albrecht Chapter 6 pp 75 – 80

3. Objectives Be able to write the vertical equation of motion
Be able to state the assumptions made for hydrostatic balance
Be able to describe hydrostatic balance

4. Objectives
Be able to derive the hydrostatic equation from the vertical equation of motion
Be able to provide the definition of geopotential
Be able to calculate geopotential height given geopotential

5. Objectives
Be able to perform calculations using the hypsometric equation
Be able to describe the relationship between average temperature in a layer and geopotential height
Be able to describe the analysis performed on constant pressure charts

6. Objectives
Be able to describe the relationship between pressure and height on constant pressure charts
Be able to explain the reason for the slope of pressure surfaces from the equator to the poles

7. Objectives Be able to provide the definition of thickness
Be able to describe the relationship between average temperature in a layer and thickness

8. Objectives
Be able to calculate thickness given the average temperature of a layer
Be able to perform calculate the average temperature of a layer given the thickness

9. Vertical Equation of Motion
Which forces are most important in the vertical?
Coriolis
Friction
Pressure Gradient
Gravity

10. Coriolis Force
Mostly Horizontal
PGF NP Co

11. Frictional Forces
Friction
Mostly Horizontal
Friction
Wind

12. Pressure Gradient
Change in pressure over a given distance dp dx

13. Pressure Gradient
Three Dimensional Pressure Gradient

14. Pressure Gradient
Let’s evaluate the vertical & horizontal pressure gradient
8 mb
150 mi
500 mb
3 mi

15. Pressure Gradient Force
Vertical Pressure
Cartesian (z) z

16. Gravity
Gravity
Ability of objects to attract each other
Gravity

17. Gravity ME Gravitational Force m Function of mass of each object
Inversely proportional to distance r ME

18. Gravity Gravitational Acceleration (g) G = 6.67 x 10-11 Nm2kg-2
Varies with
Mass
Radius
Earth’s
Height Above Ground

19. Gravity Gravitational Acceleration Assumed Constant
g = 9.8 meters/sec2
= 32 ft/sec2
Variation must be accounted

20. Equation of Motion
Vertical Equation of Motion

21. Equation of Motion Vertical Acceleration
Important Consideration in Thunderstorms

22. Equation of Motion Vertical Acceleration
Not So Important in Synoptic Meteorology

23. Vertical Equation of Motion
Pressure
Gradient
Gravity
Only Two Forces
Vertical Pressure Gradient
Gravity

24. Vertical Equation of Motion
Pressure
Gradient
Gravity
Vertical Pressure Gradient is equal to Gravity!

25. Vertical Equation of Motion
The Vertical Pressure Gradient is Balanced by Gravity! dp dz
Air
Parcel z g

26. Hydrostatic Equation
This relationship is known as the Hydrostatic Equation dp dz
Air
Parcel z g

27. Hydrostatic Equation Hydro - fluid Static - not moving dp Balance! dz
Air
Parcel z g

28. Hydrostatic Equation Rearrange a few terms dp = change in pressure
dz = change in height
r = density
g = gravity

29. Geopotential (F)
The potential energy of a unit mass relative to sea level
Numerically equal to the work that would be done in lifting the unit mass from sea level to the height at which the mass is located

30. Geopotential (F)
The work that must be done against the Earth’s gravitational field in order to raise a mass of 1 kg from sea level to that point
Glossary of Meteorology

31. Geopotential (F) dz g

32. Geopotential (F)
g = f(z) dz g

33. Geopotential Height (Z)
g = f(z)
go = 9.8ms-2
The height of a given point in the atmosphere in units proportional to the potential energy of unit mass (geopotential) at this height relative to sea level
Glossary of Meteorology

34. Geopotential Height (Z)
g = f(z)
go = 9.8ms-2
Used in upper air calculations
Small difference between height (z) and geopotential height (Z) in lower atmosphere

35. Hydrostatic Equation r We don’t normally measure density.
I’m too young to die!
Eliminate density. r

36. Hydrostatic Equation Using the Ideal Gas Law
Substitute into Hydrostatic Equation

37. Hydrostatic Equation
Rearrange terms

38. Hydrostatic Equation
Remember ...
Substitute

39. Hydrostatic Equation
Integrate between two pressure levels

40. Hydrostatic Equation
Divide both sides by go and reverse the limits

41. Hydrostatic Equation
Remember Substitute!

42. Hypsometric Equation
The height difference between two pressure surfaces depends on
Virtual Temperature
Pressure

43. Hypsometric Equation
p1 = sea level pressure
Z1 = 0 m
At Sea Level

44. Hypsometric Equation Height of a pressure surface Z2 p2= 700 mb
p1= SLP

45. Constant Pressure Chart
Height of a pressure surface

46. Constant Pressure Chart
Contours - lines of constant height

47. Constant Pressure Chart
Height of a pressure surface
Function of Temperature
700 mb H L
z = 3120 m
z = 2850 m SL

48. Constant Pressure Chart
How does height compare to pressure? H L
690 mb z p
700 mb
710 mb
720 mb
10,000 ft SL

49. Constant Pressure Chart
690 mb
700 mb L H p
710 mb
10,000 ft
720 mb SL

50. Constant Pressure Chart
High Height
High Pressure
Low Height
Low Pressure
700 mb H L
z = 3120 m
z = 2850 m SL

51. Hypsometric Equation Hypsometric Equation
Relates the distance between pressure surfaces z2
p2= 500 mb
5480 m z1
p1= 1000 mb
60 m

52. Thickness Thickness (DZ) Distance between pressure surfaces z2
p2= 500 mb
dp = p2 - p1
DZ = Z2 - Z1
5480 m z1
p1= 1000 mb
60 m

53. Thickness Calculate Thickness Problem Temperature varies with height
Integrate!?
Problem
Temperature varies with height

54. Thickness
What are we going to do?

55. Thickness
Two methods
1.) New-Miracle Analysis
2.) Fudge

56. Thickness
Take the average temperature of the layer

57. Thickness Substitute average temperature Mathematically .... where
weighted
average

58. Thickness
Simplify

59. Hypsometric Equation DZ = height between pressure surfaces
p1 = lower pressure surface
p2 = upper pressure surface
Tv = average temperature in layer
Rd = Gas Constant
go = gravity

60. Hypsometric Equation
Pressure decreases logarithmically

61. Hypsometric Equation
Pressure decreases logarithmically

62. Hypsometric Equation
Thickness of a layer depends on temperature

63. Hypsometric Equation Thickness depends on temperature 500 mb DZ DZ
Warm DZ
Cold
1000 mb

64. Hypsometric Equation Thickness depends on temperature South North Warm
500 mb
400 mb
300 mb
600 mb
700 mb
South
North
Warm
Air
Cold
Decreasing
Pressure

65. Hypsometric Equation

66. Thickness
mb common

67. Thickness